Stable rank of group C*-algebras

Specialeforsvar: Viktoria Woergaard Løgstrup

Titel: Stable rank of group C*-algebras

Abstract:  The concept of stable rank for a unital C-algebra was introduced by Rieffel in 1982. The case of stable rank one has been widely studied and is known to have interesting consequences for the C*-algebra, especially for the K-theory of the C*-algebra. This thesis begins by introducing the concept of stable rank for unital C*-algebras before moving on to the special case of stable rank one. The main focus of this thesis is to present results proving that the reduced group C*-algebra C* r (G) for certain discrete groups has stable rank one. The main result discussed is that of Dykema, Haagerup, and Rørdam which shows that C* r (G₁* G₂) has stable rank one for jG₁j  2 and jG₂j  3. The proofs use the reduced free product, a concept introduced by Voiculescu during his work with free probability theory, which is considered non-commutative probability theory. This thesis
introduces and proves the existence of the reduced free product of a family (Ai; 'i)i2I of unital C*-algebras Ai each equipped with faithful state 'i. Finally, we discuss more recent results of Gerasimova and Osin, who showed that C* _r (G) has stable rank one for a class of acylindrically hyperbolic groups.

Vejleder:  Mikael Rørdam
Censor:    Jens Kaad, SDU